An Introduction to Nonharmonic Fourier Series (Pure and by Robert M. Young

By Robert M. Young

An creation to Non-Harmonic Fourier sequence, Revised variation is an replace of a well known and hugely revered vintage textbook.Throughout the ebook, fabric has additionally been extra on fresh advancements, together with balance idea, the body radius, and functions to sign research and the regulate of partial differential equations.

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Extra resources for An Introduction to Nonharmonic Fourier Series (Pure and Applied Mathematics (Academic Pr))

Example text

Form an orthonormal basis. The asymptotic formula for u,, shows that c IIen - and the result follows at once from Theorem 16. f in L Z [ a , h ]can be expressed in terms of the eigenfunctions ; the corresponding series is called a Sturm Liouoille series. 48 Bases in Banach Spaces [Ch. 1 We conclude our discussion of stability by further clarifying the relation between Riesz bases and the Paley-Wiener criterion. As always, H is a separable Hilbert space and {ell}an orthonormal basis for H . The Paley-Wiener criterion is nothing more than the assertion that the mapping T:e,+f, for n = 1,2,3,..

5 ) : Let {g,} be the unique sequence in H biorthogonal to { J ; , ) . By Theorem 8, { g,} is also a Riesz basis for H . Since every vector f ‘ in H has the two biorthogonal expansions the result follows immediately from the definition of a Riesz basis. Sec. s] 35 Riesz Bases ( 5 ) + (1): Consider the linear transformation from H into l2 defined by The reader will verify without great difficulty that this mapping is closed. By the closed graph theorem it is continuous, and hence there exists a positive constant C for which Similarly, there exists a positive constant D for which Fix an arbitrary orthonormal basis {en) for H , and define operators S and T on the linear subspaces spanned by the sequences {f,,} and {g,,}, respectively, by setting ciei and T By virtue of the two inequalities above, we have and Since both sequences {f,)and {g,,} are complete, each of the operators S and T can be extended by continuity to a bounded linear operator on the entire space.

Accordingly, every function f in L2[-n,n] will have a unique nonharmonic Fourier series expansion 'w f (t)= C c,eianf (in the mean), -m with x l c n l 2 < co. 1 nonharmonic expansions was I iscovered by Paley and Wiener [1934], and it was for this purpose that they formulated the criterion of Theorem 13. In the present setting that criterion takes the form whenever x l c n 1 2 5 1. When shall the sequence {A, - n } be considered "small"? Based on what has already been established, one might well suppose that the condition A,,-n+O as n - + + c o is, at the very least, necessary.

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